I was studying identical particles in Quantum Mechanics, when I came across the notion of the 'exchange operator' acting on a two-particle wavefunction, $\psi_(x_1, x_2)$, in one dimension: $$ P_{12}\,\psi(x_1, x_2) = \psi(x_2, x_1)$$ The way I understand, all that $P_{12}$ does is switch the positions of the two particles. I then read about the two eigenstates of this operator, which are 'symmetric' and 'antisymmetric' and correspond to bosons and fermions, respectively: $$\psi_S(x_1, x_2) = \frac{1}{\sqrt{2}} \left(\psi(x_1, x_2) + \psi(x_2, x_1) \right) \to \text{Bosons};$$
$$\psi_A(x_1, x_2) = \frac{1}{\sqrt{2}} \left(\psi(x_1, x_2) - \psi(x_2, x_1) \right) \to \text{Fermions}.$$
My question is, what is so special about the eigenstates of this operator and why do they correspond to particles?
I have encountered this operator-particle idea in other areas as well. In Particle Physics, certain particles are described by the charge conjugation operator, for instance, which reverses the charges of particles. The symmetric and the antisymmetric eigenfunctions of the charge conjugation operator correspond to two different kinds of particles, much like in the case of bosons and fermions.
I suppose I can generalize my question to this: Does the existence of an eigenstate imply the existence of a particle, and/or vice-versa? If yes, why -- what is so special about eigenstates in particular? What about other states? Why only the eigenstates of certain operators? Thanks for your time.